Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.

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Caution: some programs are rewarding. Others lead
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to switch.

Are you a careful reader, writer and thinker?Five logic chapters lead to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. -
1 versus 2-way implication rules - A different starting point - Writing or introducting
the 1-way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
-
Deductive Chains of Reason - See which implications can and cannot be used together
to arrive at more implications or conclusions,
-
Mathematical Induction - a light romantic view that becomes serious. -
Responsibility Arguments - his, hers or no one's -
Islands and Divisions of Knowledge - a model for many arts and
disciplines including mathematics course design: Different entry
points may make learning and teaching easier. Are you ready for them?

Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. -
Decimals for Tutors - lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals. -
Primes Factors - Efficient fraction skills and later studies of
polynomials depend on this. -
Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
fractions. -
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.

What is
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
correct? -
Formula Evaluation - Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign. -
Solve
Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
equations algebraically. -
Function notation for Computation Rules - another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function? -
Axioms [some] as equivalent Computation Rule view - another way for understanding
and explaining axioms. -
Using
Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.

The computation rules labelled with odd and even subscripts
provide computation rules to describe or list odd and even numbers.
If m is an even number then long division or its decimal form implies m = 2n.
while if m is an even number then long division or its decimal form m = 2n+1.

For a second example the odd number
$109 =
= 54 \ times 2 + 1$ has the form $2n+1$ where $n=54$ It has also has the form
$2n-1$ when $n=55$

Evaluation Exercises

Calculate - that is obtain exact arithmetic expressions for the
following. Do not use a calculator.

$a_1= f(5)$

$a_2 = g(10)$

$a_3= h(2)$

$a_4= k(4,10)$

$a_5= p(2,8)$

$a_6= q(3, 4, 2)$

$a_7= r(1, 2, 3)$

$b_0=f_{even}(0)$

$b_1=f_{even}(1)$

$b_2=f_{even}(2)$

$b_3=f_{even}(3)$

$b_4=f_{even}(4)$

$A= A_1(1)$

$c_0=f_{odd}(0)$

$c_1=f_{odd}(1)$

$c_2=f_{odd}(2)$

$c_3=f_{odd}(3)$

$c_4= f_{odd}(4)$

$c_5=f_{odd}(5)$

$d_1=F_{odd}(0)$

$d_1=F_{odd}(1)$

$d_1=F_{odd}(2)$

$d_1=F_{odd}(3)$

$d_1=F_{odd}(4)$

$d_1=F_{odd}(5)$

$A_{last}= A_2(3 \mbox{ cm}, 2 \mbox{ cm}) $

Solution to four exercises follow.

The bar notation $\big|_{x=5}$ as the operation: evaluated at x =5. It
use allows one to give the formula for a function in line. Here
$f(5)$ and
$f(x)\big|_{x=5}$ have the same meaning. Writing one or the other in your solutions.
Showing both is redundant.

Remark: Bar notation is employed in calculus with
the convention
\[ F(x)\big|_a^b = F(b)-F(a) \]
as a shorthand notation for the difference F(b) - F(a). The bar notation introduced
above, a variant of the calculus use, is a method to record the formula for a function
or computation rule in the evaluation process. Doing so agrees with the site
views solution steps
should show all work, so that the doer and fellow students or teachers can observe
what has been done: Skill has to be seen to be believed. Work done well and shown
in full shows skill.

Play with this [unsigned]
Complex Number Java Applet
to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A,
Pattern Based Reason, describes
origins, benefits and limits of rule- and pattern-based reason and decisions
in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not
reach it. Online postscripts offer
a story-telling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - Magellan, the McKinley
Internet Directory:

Mathphobics, this site may ease your fears of the subject, perhaps even
help you enjoy it. The tone of the little lessons and "appetizers" on
math and logic is unintimidating, sometimes funny and very clear. There
are a number of different angles offered, and you do not need to follow
any linear lesson plan. Just pick and peck. The site also offers some
reflections on teaching, so that teachers can not only use the site as
part of their lesson, but also learn from it.

... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem

Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.

... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...

-
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions. -
Complex Numbers - Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trig-formulas for dot- and
cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no
knowledge and some knowledge of the tangent function in
trigonometry.

Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals. -
Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima
and minima while indicating why we calculate derivatives or slopes to linear and nonlinear
curves y =f(x) -
Arithmetic Exercises with hints of algebra. - Answers are given. If there are many
differences between your answers and those online, hire a tutor, one
has done very well in a full year of calculus to correct your work. You may be worse than you think.